Re: AlgebraicRules

*To*: mathgroup at smc.vnet.net*Subject*: [mg127485] Re: AlgebraicRules*From*: Andrzej Kozlowski <akozlowski at gmail.com>*Date*: Sun, 29 Jul 2012 03:03:01 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net*References*: <20120728063859.B94706847@smc.vnet.net>

Yes, you can use it in exactly the same way but AlgebraicRules is a lot more compact and does certain things for you. (On the other hand it's much harder to know what is gong on). Here are two equivalent examples: Using AlgebraicRules: Sin[x]^3 + Cos[x]^2 /. AlgebraicRules[{Sin[x]^2 + Cos[x]^2 == 1, TrigExpand[Sin[3*x]] == 1/2}] Sin[x]^2 + (3*Sin[x])/4 + 7/8 Using PolynomialReduce and GroebnerBasis: Last[ PolynomialReduce[Sin[x]^3 + Cos[x]^2, GroebnerBasis[{Sin[x]^2 + Cos[x]^2 - 1, -Sin[x]^3 + 3*Sin[x]*Cos[x]^2 - 1/2}, {Cos[x], Sin[x]}], {Cos[x], Sin[x]}]] -Sin[x]^2 + (3*Sin[x])/4 + 7/8 Note that if you omit GroebnerBasis you will get a different answer: Last[ PolynomialReduce[ Sin[x]^3 + Cos[x]^2, {Sin[x]^2 + Cos[x]^2 - 1, -Sin[x]^3 + 3*Sin[x]*Cos[x]^2 - 1/2}, {Cos[x], Sin[x]}]] Sin[x]^3 - Sin[x]^2 + 1 Under the assumption Sin[3x]==1/2 both answers are equivalent: FullSimplify[-Sin[x]^2 + (3*Sin[x])/4 + 7/8 == Sin[x]^3 - Sin[x]^2 + 1] 2*Sin[3*x] == 1 Andrzej Kozlowski On 28 Jul 2012, at 08:38, Murray Eisenberg wrote: > In a very recent post by Fred Simons, he cited his paper "Computer > algebra in service courses", available as: > > http://alexandria.tue.nl/openaccess/Metis217845.pdf > > In it, he proves a certain trig identity. He begins by using TrigExpand > to obtain a certain polynoial, call it "expanded", in Sin[x] and Cos[x]. > Then he uses the function AlgebraicRules: > > expanded/.AlgebraicRules[{ > Sin[x]^2 + Cos[x]^2 == 1, TrigExpand[Sin[3x]] == 1/2}] > > I only vaguely recall having seen AlgebraicRules back in Version 2.2 but > have not come across it since. > > Although AlgebraicRules persists in the current version of Mathematica, > the docs say that since Version 3.0, Algebraic Rules has been superseded > by PolynomialReduce. > > Can PolynomialReduce in fact be used _directly_ on an expression that is > not a polynomial in, say, a single variable x but rather in the pair of > functions Sin[x] and Cos[x]? > > Or must one revert to the artifice of replacing Sin[x] and Cos[x] by new > variable names, use PolynomialReduce, and finally reverse the > replacement to get back to the original variable x? > > -- > Murray Eisenberg murray at math.umass.edu > Mathematics & Statistics Dept. > Lederle Graduate Research Tower phone 413 549-1020 (H) > University of Massachusetts 413 545-2859 (W) > 710 North Pleasant Street fax 413 545-1801 > Amherst, MA 01003-9305 >

**References**:**AlgebraicRules***From:*Murray Eisenberg <murray@math.umass.edu>